Lambert W function是一種特殊的數學函數，近年來被應用於針對時間延遲系統之分析與控制器設計。此方法無需將延遲造成之非線性現象透過線性化近似，可直接對時間延遲系統進行分析，並以特徵值分佈的方式來推導設計出控制器參數。本論文針對文獻中所公開的MATLAB語言之工具軟體，將實現此演算法所開發的Lambert W_DDE toolbox進行使用上的說明以及設定修改，使之能順利應用在高階系統上。本文將使用旋轉倒單擺控制平台進行實驗驗證，實驗系統之時間延遲乃是為評估延遲控制之效果而刻意地人工調整，再經用Lambert W function直接對此延遲系統進行控制器設計，以實驗方式驗證此方法在現實系統中的應用可行性與困難，並與傳統控制方法做比較，討論彼此的優缺點。

Lambert W function is a special mathematic function which has been used for the analyses and control designs for the time delay systems in recent years. This method does not require the delay to be linearized and approximated, and can analyze and design the controller analytically by evaluating the system’s desired eigenvalues. In this research, a publically available MATLAB toolbox that has been developed to aid the Lambert W function control design procedures is employed and investigated for the applications of Lambert W function based control to high order systems. The procedures are detailed and the toolbox is modified accordingly. A rotary inverted pendulum system is employed as the experimental platform to evaluate the performance of the derived control laws. The time delay of the control hardware is manually adjusted purposely to evaluate the feasibility and performance of the Lambert W function based control laws. The effectiveness of the Lambert W function based method is compared with other common control methods for the systems with time delay.

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